Can a convolutional neural network classify text document images?
Yes, it is possible, CNNs can also be used for OCR (see the MNIST task and this blog), although it's not the common way for OCR because it is considered a bit overkill and inefficient. Furthermore, OCR is considered a solved problem already and don't need deep learning to do well, unless perhaps under unfavorable conditions like dark lighting, complex background of the document, weird fonts, occlusion of text, etc.
Breaking down your questions:

a CNN can likely classify an image as &quot;this image has a word document in it&quot; vs &quot;this image does not have a word document in it&quot;. The word document itself, and the outline of the words, should form distinct enough features for the CNN to discriminate against
a CNN can likely identify the vendor based on the logo of the vendor if you have enough training images of it at various distortions (angles, occlusions, different lighting, etc)
a CNN can likely do OCR well enough to extract the text, and then an entity extraction model can identify the vendor from the words. However, this approach would be uncommon as OCR is already considered a solved problem. But a neural network can potentially help preprocessing difficult OCR images (rotating image, identifying text boundaries, fix lighting, perform super-resolution, etc) to increase OCR accuracy